On positive solutions for a class of singular quasilinear elliptic systems

Abstract

We study through the lower and upper-solution method, the existence of positive weak solution to the quasilinear elliptic system with weights ⎧ ⎪ −div(|x|−ap |∇u|p−2 ∇u) = λ|x|−(a+1)p+c1 uα v γ in Ω, ⎨ −div(|x|−bq |∇v|q−2 ∇v) = λ|x|−(b+1)q+c2 uδ v β in Ω, ⎪ ⎩ u=v=0 on ∂Ω, −p −q where Ω is a bounded smooth domain of RN , with 0 ∈ Ω, 1 < p, q < N , 0 a < N p , 0 b < N q , 0 α < p − 1, 0 β < q − 1, δ, γ , c1 , c2 > 0 and θ := (p − 1 − α)(q − 1 − β) − γ δ > 0, for each λ > 0.